Question

The force required to compress a non-standard spring as a function of displacement is given by the equation?

Answer 1

Work done by a spring is calculated by integrating the spring force from one position to another. For the given spring force function F(x) and the defined constants A, b, and k, we find the general work equation and then calculate specific work done over intervals 0 to 0.21 m and 0.21 m to 0.51 m.

Step-by-step explanation:

The calculation of work done by a spring involves integrating the force over the distance of compression or extension. Given the force function F(x) = -Asin(bx) + kx and the constants A = 11 N, b = 7 rad/m, and k = 42 N/m, we calculate:

1. The general equation for work from x1 to x2: W = ∫ (F(x) dx) which becomes W = ∫ [-Asin(bx) + kx] dx.
2. Calculate the work done from x = 0 to x1 = 0.21 m
3. Calculate the work done from x1 = 0.21 m to x2 = 0.51 m

These integrations can be done using standard calculus methods, such as integration by parts for the sinusoidal term and direct integration for the linear term.